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Impact of inhomogeneous fiber-reinforced layer with frictional interface on Rayleigh-type wave propagation

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Abstract

The effect of frictional boundary on the propagation of Rayleigh-type wave in an initially stressed inhomogeneous fiber-reinforced layer overlying an initially stressed homogeneous semi-infinite medium has been analyzed by an approximate analytical method. A realistic model has been considered for sliding boundary friction at the interface. The frequency equation has been obtained in closed form. The substantial effects of various affecting parameters, viz. reinforcement, inhomogeneity, bonding parameter, spectral decay parameter, and horizontal initial stress on phase and damped velocity have been discussed graphically in detail. The remarkable observation has been obtained through the comparative study in the presence and the absence of reinforcement in the layer.

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Acknowledgements

The authors convey their sincere thanks to the Indian Institute of Technology (ISM), Dhanbad for providing JRF to Ms. Akanksha Srivastava and also facilitating us with its best facility for research.

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  1. Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Akanksha Srivastava, Amares Chattopadhyay & Abhishek Kumar Singh

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  1. Akanksha Srivastava
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  2. Amares Chattopadhyay
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  3. Abhishek Kumar Singh
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Correspondence to Akanksha Srivastava.

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Appendices

Appendix A

$$\begin{aligned} A_{11}&=-n^{2} \left( {\frac{Q}{\mu _{\mathrm{T}1}^{(0)}}-\frac{P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}} \right) +\mathrm{i}n \left( {\frac{-2\mathrm{i}N}{\mu _{\mathrm{T}1}^{(0)}}+\frac{\gamma '}{\varepsilon } \frac{Q}{\mu _{\mathrm{T}1}^{(0)}}- \frac{\gamma '}{\varepsilon } \frac{P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}} \right) + \left( {\frac{-L}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i} \gamma '}{\varepsilon } \frac{N}{\mu _{\mathrm{T}1}^{(0)}}+\frac{\rho _{1}c^{2} \eta {2}}{\mu _{\mathrm{T}1}^{(0)}}} \right) ,\\ A_{12}&=\frac{-J}{\mu _{\mathrm{T}1}^{(0)}} n^{2}+ \mathrm{i}n\left( \frac{\gamma '}{\varepsilon } \frac{J}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}M}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}Q}{\mu _{\mathrm{T}1}^{(0)}} \right) + \left( {\frac{-N}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}Q}{\mu _{\mathrm{T}1}^{(0)}} \frac{\gamma '}{\varepsilon } +\mathrm{i}\frac{\gamma '}{\varepsilon }\frac{P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}} \right) , \\ A_{21}&=\frac{-J}{\mu _{\mathrm{T}1}^{(0)}} n^{2}+ \mathrm{i}n\left( \frac{\gamma '}{\varepsilon } \frac{J}{\mu _{\mathrm{T}1}^{(0)}}+\frac{\mathrm{i}P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}M}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}Q}{\mu _{\mathrm{T}1}^{(0)}} \right) + \left( {\frac{-N}{\mu _{\mathrm{T}1}^{(0)}}-\frac{\mathrm{i}M}{\mu _{\mathrm{T}1}^{(0)}} \frac{\gamma '}{\varepsilon } } \right) , \\ A_{22}&= \frac{R}{\mu _{\mathrm{T}1}^{(0)}} n^{2}+\mathrm{i}n \left( {\frac{\gamma '}{\varepsilon } \frac{R}{\mu _{\mathrm{T}1}^{(0)}} -\frac{-2\mathrm{i}J}{\mu _{\mathrm{T}1}^{(0)}} } \right) + \left( {\frac{-Q}{\mu _{\mathrm{T}1}^{(0)}} - \frac{\mathrm{i}J}{\mu _{\mathrm{T}1}^{(0)}}\frac{\gamma '}{\varepsilon }-\frac{P_{1}}{2\mu _{\mathrm{T}1}^{(0)}}+\frac{\rho _{1}c^{2} \eta {2}}{\mu _{\mathrm{T}1}^{(0)}}} \right) ,\quad \eta =(1+\mathrm{i}\kappa ),\quad \beta _{1}^{2}=\frac{{\mu _{\mathrm{T}1}^{(0)}}}{\rho _{1}^{(0)}},\\ b_{1}&= \left[ { - {{\left( {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) }^2} + \frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] ,\quad {b_2} = 2\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \mathrm{i}\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) \\&\quad - \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) \left( {\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} - 2\mathrm{i}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) - \left( {\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) \left[ { - 2\mathrm{i}\left( {\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) + \left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) - \left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right] ,\\ {b_3}&= \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left[ { - \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] + \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left[ { - \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) + \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] \\&\quad + \left( {\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) \left[ { - \frac{L}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] - \left[ {\left( {\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) \left( {\frac{{\gamma '}}{\varepsilon }} \right) - \frac{{2J}}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] \\&\quad \times \left[ { - 2\mathrm{i}\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} + \left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) - \left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] \\&\quad - \left[ { - \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) - \mathrm{i}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \left( {\frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] \left[ {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right] \\&\quad + \left[ {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i} - \frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i} - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] \left[ {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i} - \frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i} + \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] ,\\ {b_4}&= \left[ {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - i\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{M}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] \left[ {\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} + \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] \\&\quad - \left[ { - \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) + \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] \left[ {\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \mathrm{i}\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{M}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] \\&\quad + \left[ {\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} - 2\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\mathrm{i}} \right] \left[ { - \frac{L}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] \\&\quad + \left[ { - 2\mathrm{i}\left( {\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) + \left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) - \left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right] \left[ { - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] ,\\ {b_5}&= \left[ { - \frac{L}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] \left[ { - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}\left( {\frac{{\gamma '}}{\varepsilon }} \right) - \frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{\eta ^2}} \right] \\&\quad + \left[ {\frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} + \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{M}{{\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] \left[ { - \frac{N}{{\mu _{\mathrm{T}1}^{(0)}}} - \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}} + \mathrm{i}\left( {\frac{{\gamma '}}{\varepsilon }} \right) \left( {\frac{{P_0^{(1)}}}{{2\mu _{\mathrm{T}1}^{(0)}}}} \right) } \right] . \end{aligned}$$

Appendix B

$$\begin{aligned} d_{1}= & {} \left( {1 - \frac{{{P_2}}}{{2{\mu _2}}}} \right) \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) ,\nonumber \\ {d_2}= & {} \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 1 + \frac{{{P_2}}}{{2{\mu _2}}}} \right) \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 1 - \frac{{{P_2}}}{{2{\mu _2}}}} \right) + \left( {1 - \frac{{{P_2}}}{{2{\mu _2}}}} \right) \left( {\frac{{{v^2}{\eta ^2}}}{{{\beta ^2}}} - \left( {1 + \frac{{{P_2}}}{{2{\mu _2}}}} \right) } \right) \nonumber \\&+ \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) \left( {\frac{{{v^2}{\eta ^2}}}{{{\beta ^2}}} - \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) } \right) ,\nonumber \\ {d_3}= & {} \left( {\frac{{{v^2}{\eta ^2}}}{{{\beta ^2}}} - \left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) } \right) \left( {\frac{{{v^2}{\eta ^2}}}{{{\beta ^2}}} - \left( {1 + \frac{{{P_2}}}{{2{\mu _2}}}} \right) } \right) . \end{aligned}$$

Appendix C

$$\begin{aligned}&{a_{11}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _1} + {{n_1}}{\delta _1}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{ - \mathrm{i}k{{n_1}}H}},\nonumber \\&{a_{12}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _2} + {{n_2}}{\delta _2}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{ - \mathrm{i}k{{n_2}}H}},\\&{a_{13}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _1} - {{n_1}}{\chi _1}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{\mathrm{i}k{{n_1}}H}},\nonumber \\&{a_{14}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _2} - {{n_2}}{\chi _2}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{\mathrm{i}k{{n_2}}H}},\\&{a_{15}} = 0, \quad {a_{16}} = 0,\quad {a_{21}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _1} + {{n_1}}{\delta _1}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{ - \mathrm{i}k{{n_1}}H}},\\&{a_{22}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _2} + {{n_2}}{\delta _2}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{ - \mathrm{i}k{{n_2}}H}},\nonumber \\&{a_{23}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _1} - {{n_1}}{\chi _1}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{\mathrm{i}k{{n_1}}H}},\\&{a_{24}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _2} - {{n_2}}{\chi _2}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}}} \right] {\mathrm{e}^{\mathrm{i}k{{n_2}}H}},\quad {a_{25}} = 0,\quad {a_{26}} = 0,\\&{a_{31}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _1} + {{n_1}}{\delta _1}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}} + \phi } \right] ,\\&{a_{32}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _2} + {{n_2}}{\delta _2}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}} + \phi } \right] ,\quad {a_{35}} = - \phi ,\\&{a_{33}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _1} - {{n_1}}{\chi _1}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}} + \phi } \right] ,\\&{a_{34}} = \left[ {\frac{{ - N}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{Q}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _2} - {{n_2}}{\chi _2}\frac{J}{{\mu _{\mathrm{T}1}^{(0)}}} + \phi } \right] ,\quad {a_{36}} = - \phi , \end{aligned}$$
$$\begin{aligned}&{a_{41}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _1} + {{n_1}}{\delta _1}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} + \psi {\delta _1}} \right] ,\nonumber \\&{a_{42}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} + \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\delta _2} + {{n_2}}{\delta _2}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} + \psi {\delta _2}} \right] ,\\&{a_{43}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_1}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _1} - {{n_1}}{\chi _1}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} + \psi {\chi _1}} \right] ,\\&{a_{44}} = \left[ {\frac{{ - M}}{{\mu _{\mathrm{T}1}^{(0)}}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{{n_2}} - \frac{J}{{\mu _{\mathrm{T}1}^{(0)}}}{\chi _2} - {{n_2}}{\chi _2}\frac{R}{{\mu _{\mathrm{T}1}^{(0)}}} + \psi {\chi _2}} \right] ,\\&{a_{45}} = - \psi {\ell _3},\quad {a_{46}} = - \psi {\ell _4},\quad {a_{51}} = \frac{\phi }{\mu },\quad {a_{52}} = \frac{\phi }{\mu },\quad {a_{53}} = \frac{\phi }{\mu },\quad {a_{54}} = \frac{\phi }{\mu },\quad \psi = v\left( {( {1 + \mathrm{i}\kappa })\overline{{M_2}} - i\overline{{L_2}} } \right) ,\\&{a_{55}} = - \left( {{{m_1}} + {\ell _3} + \frac{\phi }{\mu }} \right) ,\quad {a_{56}} = - \left( {{{m_2}} + {\ell _4} + \frac{\phi }{\mu }} \right) ,\quad {a_{61}} = \frac{\psi }{\mu },\quad {a_{62}} = \frac{\psi }{\mu },\quad {a_{63}} = \frac{\psi }{\mu },\quad {a_{64}} = \frac{\psi }{\mu },\\&{a_{65}} = - \left( {{{m_1}}{\ell _3}\left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) + \frac{\psi }{\mu }{\ell _3} + \frac{{{\lambda _2}}}{{{\mu _2}}}} \right) ,\quad {a_{66}} = - \left( {{{m_2}}{\ell _4}\left( {\frac{{{\lambda _2}}}{{{\mu _2}}} + 2} \right) + \frac{\psi }{\mu }{\ell _4} + \frac{{{\lambda _2}}}{{{\mu _2}}}} \right) ,\\&\phi = v\left( {( {1 +\mathrm{i}\kappa } )\overline{{M_1}} - \mathrm{i}\overline{{L_1}} } \right) . \end{aligned}$$

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Srivastava, A., Chattopadhyay, A. & Singh, A.K. Impact of inhomogeneous fiber-reinforced layer with frictional interface on Rayleigh-type wave propagation. J Eng Math 114, 159–176 (2019). https://doi.org/10.1007/s10665-019-09987-5

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